Just because you're a part of the fandom doesn't mean you have to flaunt it with public furry wear.
If it's uncomfortable for you then you shouldn't force yourself to do it.
You have to keep in mind that by wearing furry gear outside of furry socials you will get looked at; people will stare and maybe even laugh, maybe something worse. I'm not trying to be discouraging, but it's something you have to learn to expect if you want to do that.
In the end it's up to you, but you don't really gain furry points or get some higher ranking by being public, so it's not a necessity, but by all means if you want to express yourself no one is stopping you. You just have to keep in mind the consequences; the world isn't exactly filled with cupcakes and flowers all the time.